Abstract

We define Galilean invariant delta functions by using the Takahashi formulation, which describes particles with arbitrary spin. The direct calculation of the Galilean invariant delta functions in the -dimensional Galilean covariant extended manifolds reveals that, for odd, there exists a surface on which these delta functions vanish. This feature corresponds to the spacelike separation in the relativistic quantum field theory, and it enables us to discuss a connection between spin and statistics in the Galilean covariant theories. These results show that a spin-statistics connection holds for even-dimensional Galilean covariant extended manifolds; that is, for -dimensional space-times with odd . Likewise, we observe that there is no such connection for odd-dimensional Galilean covariant extended manifolds; that is, for -dimensional space-times with even . Thus, our results support the claim that there is no connection between spin and statistics in the usual nonrelativistic theory, for which .